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Piecewiselinear models of genetic regulatory networks: theory and example
 IN BIOLOGY AND CONTROL THEORY: CURRENT CHALLENGES, LECTURE
"... The experimental study of genetic regulatory networks has made tremendous progress in recent years resulting in a huge amount of data on the molecular interactions in model organisms. It is therefore not possible anymore to intuitively understand how the genes and interactions together influence th ..."
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The experimental study of genetic regulatory networks has made tremendous progress in recent years resulting in a huge amount of data on the molecular interactions in model organisms. It is therefore not possible anymore to intuitively understand how the genes and interactions together influence the behavior of the system. In order to answer such questions, a rigorous modeling and analysis approach is necessary. In this chapter, we present a family of such models and analysis methods enabling us to better understand the dynamics of genetic regulatory networks. We apply such methods to the network that underlies the nutritional stress response of the bacterium E. coli.
Limit cycles in piecewiseaffine gene network models with multiple interaction loops
, 2009
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Qualitative control of periodic solutions in piecewise affine systems; application to genetic networks
, 2009
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Probabilistic approach for predicting periodic orbits in piecewise affine differential models
 in "Bull. Math. Biol
"... Piecewise affine models provide a qualitative description of the dynamics of a system, and are often used to study genetic regulatory networks. The state space of a piecewise affine system is partitioned into hyperrectangles which can be represented as nodes in a directed graph, so that the system’s ..."
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Piecewise affine models provide a qualitative description of the dynamics of a system, and are often used to study genetic regulatory networks. The state space of a piecewise affine system is partitioned into hyperrectangles which can be represented as nodes in a directed graph, so that the system’s trajectories follow a path in a transition graph. This paper proposes and compares two definitions of transition probability between two nodes A and B of the graph, based on the volume of the initial conditions on the hyperrectangle A whose trajectories cross to B. The parameters of the system can thus be compared to the observed transitions between two hyperrectangles. This property may become useful to identify sets of parameters for which the system yields a desired periodic orbit with a high probability, or to predict the most likely periodic orbit given a set of parameters, as illustrated by a gene regulatory system composed of two intertwined negative loops. 1
Transition probabilities for piecewise affine models of genetic networks
, 2013
"... Abstract — In the piecewise affine framework, trajectories evolve among hyperrectangles in the state space. A qualitative description of the dynamics useful for models of genetic networks can be obtained by viewing each hyperrectangle as a node in a discrete system, so that trajectories follow a p ..."
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Abstract — In the piecewise affine framework, trajectories evolve among hyperrectangles in the state space. A qualitative description of the dynamics useful for models of genetic networks can be obtained by viewing each hyperrectangle as a node in a discrete system, so that trajectories follow a path in a transition graph. In this paper, a probabilistic interpretation is given for the transition between two nodes A and B, based on the volume of the initial conditions on hyperrectangle A whose trajectories cross to B. In an example consisting of two intertwinned negative loops, this probabilistic interpretation is used to predict the most likely periodic orbit given a set of parameters, or to find parameters such that the system yields a desired periodic orbit with a high probability. I.
Qualitative
, 2013
"... control of periodic solutions in piecewise affine models of genetic networks ..."
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control of periodic solutions in piecewise affine models of genetic networks
IN
, 2013
"... cycles in piecewiseaffine gene network models with multiple interaction loops ..."
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cycles in piecewiseaffine gene network models with multiple interaction loops
of circadian rhythms in cyanobacteria
, 2013
"... Author manuscript, published in "" This is a preliminary version of the article published as: ..."
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Author manuscript, published in "" This is a preliminary version of the article published as:
Numerical Simulation of . . . Models of Gene Regulatory Networks Using Complementarity Systems
, 2013
"... Gene regulatory networks control the response of living cells to changes in their environment. A class of piecewiselinear (PWL) models, which capture the switchlike interactions between genes by means of step functions, has been found useful for describing the dynamics of gene regulatory networks ..."
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Gene regulatory networks control the response of living cells to changes in their environment. A class of piecewiselinear (PWL) models, which capture the switchlike interactions between genes by means of step functions, has been found useful for describing the dynamics of gene regulatory networks. The step functions lead to discontinuities in the righthand side of the differential equations. This has motivated extensions of the PWL models based on differential inclusions and Filippov solutions, whose analysis requires sophisticated numerical tools. We present a method for the numerical analysis of one proposed extension, called AizermannPyatnitskii (AP)extension, by reformulating the PWL models as a mixed complementarity system (MCS). This allows the application of powerful methods developed for this class of nonsmooth dynamical systems, in particular those implemented in the Siconos platform. We also show that under a set of reasonable biological assumptions, putting constraints on the righthand side of the PWL models, APextensions and classical Filippov (F)extensions are equivalent. This means that the proposed numerical method is valid for a range of different solution concepts. We illustrate the practical interest of our approach through the numerical analysis of three wellknown networks developed in the field of synthetic biology.